Matt Van Essen

In this lecture we will review some basic mathematical concepts employed throughout the course:

1 _{Single Variable Functions}

2 Limit of a Function and Continuity

3 Derivative, Higher Order Derivatives, Di¤erential

4 Multi-Variable Functions

5 Partial Derivatives of a Multi Variable Functions

6 _{Total Di¤erential, Chain Rule, Second Partial Derivatives}

7 Simple Integration

De…nition

y is a function of x if there is a relationship between y andx that de…nes, for each value of x, a corresponding value ofy.

We describe a few special families of single variable functions:

linear functions,

general polynomial functions, the exponential function, and the natural logarithm function.

A linear function is a function of the form f(x) =ax +b, wherea

andb are parameters.

The parameterais the slope of the function and the parameterbis the vertical intercept – i.e., the value the function takes atx =0.

The main property of a linear function is that the dependent variable changes at a constant rate with respect to the independent variable – i.e., their graph is just a straight line.

Example

The graph of the linear functionf(x) =2x+1 is given below:

-5 -4 -3 -2 -1 1 2 3 4 5 -4 -2 2 4

### x

### y

Figure: Linear Function f(x) =2x+1

The linear function is a special case of a polynomial function. An n-th degree polynomial takes the form

f(x) =a0+a1x+a2x2+ +anxn.

The quadratic function – i.e., a function of the form

The graph of a quadratic is easy to remember it is either a

“mountain” or a “valley.” It looks like a mountain if a2 <0, and it

looks like a valley if a2>0. A quadratic with a2 >0 is given below.

-5 -4 -3 -2 -1 1 2 3 4 5 -4 -2 2 4

### x

### y

Figure: Example of Quadratic witha2 >0.

De…nition

Ify =f(x), then x=h(y)is the inverse function off iff(h(y)) =y.

Example

Ify =f(x) =2x, then the functionx =h(y) = 1_{2}y is the inverse function
since f(h(y)) =21_{2}y =y

The exponential and logarithm functions are two useful functions for a variety of reasons.

The exponential function is a function of the formf(x) =aex _{and}
the natural logarithm function is of the form f(x) =lnx.

-5 -4 -3 -2 -1 1 2 3 4 5 -4 -2 2 4

### x

### y

Figure: The exponential and natural logartihm functions

Logarithms and exponential functions are inverses of one another. In other words,

lnex = x

In addition to the two inverse relationships, the following four properties of these functions are also useful ones to keep in mind.

1 ln(xy) =lnx+lny
2 ln(x
y) =lnx lny
3 lnxn =nlnx
4 _{e}x+y =_{e}x_{e}y
Example

Supposef(x,y) =x2_{y}3 _{where}_{x}_{,}_{y} _{>}_{0, then by properties (1) and (3) we}

have that lnf(x,y) =2 lnx+3 lny.

In economics, we are often interested in measuring the rate of change of a function.

For instance, a …rm is interested in knowing how much pro…t will change if it produces an additional unit of output.

A manager may care about how production is changing when he adds an additional unit of labor to the production line.

A buyer wants to know how much happier they will be if the consume another unit of a certain product.

All of these questions are about marginal changes and it turns out that the set of tools provided by calculus are well suited for answering these types of questions.

A derivative tells us the instantaneous rate of change of a function.

We …rst need to …rst de…ne rate of change.

Suppose we have a function f :**R**_{!}**R**and want to know how this

function would change if we go from input x to a new input x+∆x,

where ∆x is the change in x.

In order to …nd thechange in f, denoted ∆f, we can simply take

f(x+∆x), the value of the function at the new input x+∆x, and subtractf(x), the value the function takes at the old inputx – i.e.,

∆f =f(x+∆x) f(x).

The rate of change is de…ned by ∆_{∆}f_{x} or written out

∆f

∆x =

f(x+∆x) f(x)

∆x .

Geometrically, this rate is the slope of a line that starts at the point

Figure: Rate of Change

For example, suppose a …rm was producing 10 units of output with 5 workers and that after they added 3 more workers to the production process output jumped to 12.

If we letf(l) be the production function that tells us how much

output we can produce when we usel workers, the change in output

was ∆f =f(8) f(5) =2 and the rate of change ∆_{∆}f_{l} = 2_{3}.

The derivative is the instantaneous rate of change (or the marginal change) – i.e., it is the rate of change of a function when we make a “small” change in the input. Speci…cally, the derivative is de…ned as

d

dxf(x) =∆limx_{!}0

f(x+∆x) f(x)

Graphically, the derivative of the function f at the pointx =x0 is the

slope of a line that is just tangent to f(x0). This is illustrate below.

Figure: Derivative

In summary, a derivative tells us how a function is changing at a particular point (i.e., the slope). Graphically, this is just the slope of a particular tangent line.

We care about derivatives because they tell us how the function is changing.

In particular, the sign of the derivative tells us whether the function
increasing _{dx}df >0 , decreasing df_{dx} <0 , or constant df_{dx} =0 .
The actual derivative evaluated at a point is just a number which tells
us the magnitude of the change – i.e., how fast is the function

changing. In other words, a function with a derivative equal to 10 at

x =1 is increasing faster than a function with a derivative of 2 at

Now we come to the problem of how to …nd the derivative of a function.

Of course, we have a de…nition of the derivative and we could always apply this de…nition to a particular function.

The point you should take away from the following example is that this way of calculating a derivative is rather annoying and tedious.

Suppose we want to compute the derivative of the function

f(x) =25x 1_{2}x2.

Using the de…nition of the derivative we know we have to compute ∆_{∆}f_{x}
and then take a limit of this expression as ∆x _{!}0. First, we compute
the pieces of ∆_{∆}f_{x}. The value of the functionf evaluated at the new
point x+∆x is
f(x+∆x) = 25(x+∆) 1
2(x+∆)
2
= 1
2x
2 _{x}_{(}_{∆}_{x}_{) +}_{25}_{x} 1
2(∆x)
2
+25(∆x).

The value of the function f at the original pointx is

f(x) =25x 1

2x

2_{.}

Therefore the change in f is

∆f =f(x+∆x) f(x) = 1_{2}(∆x)2 (∆x)x+25(∆x)

The change inx is just∆x.

Therefore the rate of change off is

f(x+∆x) f(x)
∆x =
25(x+∆x) 1_{2}(x+∆x)2 25x 1_{2}x2
∆x
=
1
2(∆x)
2 _{(∆}
x)x+25(∆x)
∆x
= 25 x 1
2∆x

The derivative of f is the instantaneous rate of change so

d

dxf(x) =∆limx_{!}025 x

1

2∆x =25 x.

Finally, it is useful to evaluate this derivative at several points in order to see how the function is changing at di¤erent points.

At the point x =5, the derivative off is 20 so the function is increasing. At the point 25 the derivative is zero so the function is constant. Last, at the pointx =30, the derivative is negative so the function is decreasing.

There are simple rules for calculating most derivatives.

Theorem

Suppose f(x) =a, where a is a constant real number, then the derivative of f with respect to x is

df

Theorem

Suppose f(x) =g(x) +h(x), then the derivative of f with respect to x is df dx(x) = dg dx(x) + dh dx(x).

The power rule for taking a derivative applies to polynomial functions.

Theorem

Suppose f(x) =axb, where a and b are constant real numbers, then the derivative of f with respect to x is

df

dx(x) =abx

Example

Supposef(x) =2, then _{dx}df(x) =0.

Example

Supposef(x) =3x, then _{dx}df(x) =3. Recall thatx0 =1.

Example
Supposef(x) =3x2, then _{dx}df(x) =6x.
Example
Supposef(x) =3x 12, then df
dx(x) =
3
2x
3
2.

Example

Suppose we have a polynomial function f(x) =1+x 1_{2}x2+5x3, then

df

dx(x) = 0+1 x+15x

2

Theorem Suppose f(x) =f1(x)f2(x), then df dx(x) = df1 dx(x)f2(x) +f1(x) df2 dx(x).

Example
Supposef(x) =3 lnx, then
df
dx(x) =
3
x.
Example
Supposef(x) = (3x2_{)(}_{x}4 _{x}_{+}_{1}_{)}_{, then}
df
dx(x) = (6x) (x
4 _{x}
+1) + (3x2)(4x3 1)
= 18x5 9x2+6x.

The quotient rule applies when the function we are interested in taking the derivative can be thought of as the ratio of two functions.

Theorem
Suppose f(x) = f1(x)
f2(x), then
df
dx(x) =
df1
dx(x)f2(x) f1(x)
df2
dx(x)
(f2(x))2
.
Example
f(x) = f1(x)
f2(x) =
x2
x , then
df
dx(x) =
(2x)(x) (x2_{)(}_{1}_{)}
(x)2 =
2x2 _{x}2
x2 =1

The derivative of the natural logarithm function and its inverse, the exponential function have their own rules.

Theorem

Suppose f(x) =lnx, where x >0, then

df
dx(x) =
1
x.
Theorem
Suppose f(x) =ex, then
df
dx(x) =e
x_{.}

The chain rule applies when the function we are interested in taking
the derivative can be thought of as the composition of two functions.
For example, the functionh(x) =ln(2x2_{)}_{can be thought of as the}

composition of the functions f(x) =ln(x)andg(x) =2x2 – i.e.,

h(x) =f(g(x)).
Theorem
Suppose i(x) =f(g(x)), then
di
dx(x) =
df
dx(g(x))
dg
dx(x).
Example
Supposef(x) =ln 2x2 , then _{dx}df(x) = _{2}1_{x}2 (4x) = _{x}2.

Example

(Derivative of an Inverse Function): Suppose f 1 is the inverse function of

f, then

f(f 1(x)) =x.

If we take the derivative of both sides with respect to x we have

df dx(f

1_{(}_{x}_{))}df 1

dx (x) =1.

Thus, the derivative of the inverse function f 1 is

df 1 dx (x) = 1 df dx(f 1(x)) .

Example

Supposef(x) =ex and f 1(x) =ln(x). Since the exponential and logarithm functions are inverses of one another we have that

df 1 dx (x) = d dx ln(x) = 1 df dx(f 1(x)) = 1 eln(x) = 1 x.

Similarly, if we let f(x) =ln(x)andf 1(x) =ex. Then

df 1
dx (x) =
d
dxe
x _{=} 1
1
ex
=ex.

The derivative of a function is also a function which, in some circumstances, can also be di¤erentiated.

In particular, the derivative of df_{dx}(x), which is denoted d_{dx}2f2(x) or

sometimes f00(x), is called a second order derivative.

This is the only “higher order” derivative we will need in this course. The second derivative of a function tells us how the …rst order

derivative function is changing withx. This information is very useful, for example, when trying to determine the maximum of a function.

Example

Supposef(x) = x2+10, then _{dx}df(x) = 2x. The second order
derivative is d_{dx}2f2(x) = 2. This tells us that the function

df dx(x)is decreasing at a constant rate.

Supposef(x) =lnx, wherex >0.

Then the …rst derivative off is df_{dx}(x) = 1_{x} >0 for x>0.
Thus, the logarithm function is an increasing function.
The second order derivative is d_{dx}2f2(x) =

1

x2 <0.

0 1 2 3 4 5 0 1 2 3 4 5

### x

### y

Figure: First Derivative off(x) =ln(x)

Almost every function you work with in economics is a multi-variable
real valued functionf :**R**N _{!}**R**.

Utility functions and production functions are examples of real valued multi-variable functions that will use frequently in this course.

We will brie‡y discuss how to graph 2 variable functions using three popular functions: linear function, “Cobb-Douglas” function, and the min function.

Consider the functionf :**R**2+!**R** that is de…ned by
f(x,y) =x+y

for all (x,y)_{2} **R**2

+.

This is a relatively simple linear function. It takes, as input, a point

(x,y)and adds the x component and the y component together.

below. 0 2 4

### y

10### z

0 5 0### x

2 4This 3-D depiction of the function demands too much “artistic” ability to be useful.

We will use “level curves” to graph a multi-variable function. A level curve is a 2-D way of plotting a 3-D object.

For example, a map of a mountain uses contour lines to indicate all of the spots on the mountain that have the same elevation.

A level curve is the same thing. In particular, it is a line that connects all of the (x,y)such that function f obtains the same value.

The level curves f(x,y) =1,f(x,y) =3, andf(x,y) =5. We can solve for the equation that de…nes the level curve. For instance, the level curvef(x,y) =1 is de…ned by the equation

f(x,y) =x+y =1

if we solve fory we gety =1 x.

f :**R**2 _{!}**R** of the form

f(x,y) =Axayb

with parametersA,a,b _{2}**R**+.

For example, the functionf(x,y) =x2y is a member of the

Cobb-Douglas family with A=1, a=2, and b=1. A 3-D plot of

the Cobb-Douglas functionf(x,y) =x2y is given below:

4

### y

2 0_{0}0 4 2 2

### z

4Again, to solve for a level curvef(x,y) =k we set the function equal tok. For the example, the level curves will be de…ned by the equation

x2_{y} _{=}_{k} _{or} _{y} _{=} k

x2. A few of these level curves from this same

function are: 0 1 2 3 4 5 0 1 2 3 4 5

### x

### y

Figure: Level Curves for the Cobb-Douglas Functionf(x,y) =x2y

f(x,y) =cmin_{f}ax,by_{g},
where parametersa,b,c _{2}**R**+.

The 3-D plot of this function looks like a side of a pyramid.

0 0 4 0

### y

5### x

2 5 10### z

The level curves of the minimum function look like the letter “L.”
The corners of these level curves are along the liney = _{b}ax. This
equation is found by setting the “ax” component in the min function

equal to the “by” component of the min function.

Figure: Level Curves of the Min Functionf(x,y) =min_{f}2x,y_{g}

If you know how to take derivatives of single variable functions then you know how to take partial derivatives of multi-variable functions. The trick is that we just pretend that all variables, other than one we are di¤erentiating with respect to, are constants and then proceed as before.

Example

Supposef(x,y) =x2y, then the partial derivatives with respect tox and

y are
*∂*f
*∂*x =
2xy
*∂*f
*∂*y = x
2

Supposef :**R**N _{!}**R**, then the partial derivative n
*∂*f

*∂*xi

is a function of N variables. Moreover, if this function is di¤erentiable, then we can take partial derivatives of it as well. These are higher order derivatives. They tell us how the partial derivative function is changing with respect to di¤erent variables.

We will mostly be concerned with …rst and second order derivatives of a function for a reason which we will learn about tomorrow-ish. A function of N variables has N …rst order derivatives andN2 second order derivatives.

Example

Supposef(l,k) =4l14k 3

4, then the …rst order partial derivatives are
*∂*f
*∂*l =l
3
4k
3
4 and *∂*f
*∂*k =3l
1
4k
1

4 the second order partial derivatives of f are

*∂*2f
*∂*l2 =
3
4l
7
4_{k}
3
4
*∂*2f
*∂*k*∂*l =
*∂*2f
*∂*l*∂*k =
3
4l
3
4k
1
4
*∂*2f
= 3l14k
5
4

In the example we saw that *∂*2f
*∂*k*∂*l =

*∂*2f

*∂*l*∂*k, this was not a coincidence.
Theorem (Young’s Theorem)

Suppose that y =f(x1, ...,xN)is C2 on an open region J in**R**N. Then for

all x _{2}J and for each i ,j we have that
*∂*2f
*∂*xi*∂*xj

= *∂*

2_{f}

*∂*xj*∂*xi

The partial derivative tells us how a function changes when we vary one variable and hold the others …xed. Sometimes we wish to know how a function changes when more than one variable is changed. In particular, suppose z =f(x,y) andx goes fromx1 to x2 andy

goes fromy1 to y2, then ∆z =f(x2,y2) f(x1,y1).

We can approximate ∆z by considering a plane that is tangent tof at

(x1,y1).

The changes on this plane are denoted the by total di¤erential formula

dz = *∂*f

*∂*x

dx+ *∂*f

*∂*y

dy

The total di¤erential is an approximation for∆z whenx changes by

∆x andy changes by∆y – i.e.,

Approximate _{q}
(4.1)3 (2.95)3 (1.02)3
via di¤erentials.
Well let
f(x,y,z) =px3 _{y}3 _{z}3
then
df = *∂*f
*∂*xdx+
*∂*f
*∂*ydy+
*∂*f
*∂*zdz

df = *∂*f
*∂*xdx+
*∂*f
*∂*ydy+
*∂*f
*∂*zdz
Choose (x1,y1,z1) = (4,3,1), then
*∂*f
*∂*x (x1,y1,z1) = 4
*∂*f
*∂*y (x1,y1,z1) =
9
4
*∂*f
*∂*z (x1,y1,z1) =
1
4
The di¤erential is
df = (4)(0.1) + ( 9
4)( 0.05) +
1
4 (0.02)

The di¤erential is df = (4)(0.1) + ( 9 4)( 0.05) + 1 4 (0.02) = 0.5075 Thus, ∆f =f (x2,y2,z2) f (x1,y1,z1) 0.5075 Since f (x1,y1,z1) =6, we havef (4.1,2.95,1.02) 6.5075

The actual value is 6.495 2.

What is d_{d}ln_{ln}y_{x}?
Well,
dlny = 1
ydy
dlnx = 1
xdx
Thus,
dlny
dlnx =
dy
dx
x
y

How do you …nd the change of the functionf(x,y)with respect toy

when x and y are related. Use the total derivative. Supposez =f(x,y)andx =g(y)

First, …nd the total di¤erential

dz = *∂*f

*∂*xdx+

*∂*f

*∂*ydy

Next, divide both sides by dy
dz
dy =
*∂*f
*∂*x
dx
dy +
*∂*f
*∂*y

This decomposes the change in y into a “direct e¤ect”*∂*f

*∂*y and an

“indirect e¤ect”*∂*f
*∂*x

dx dy.

Suppose

f(x,y) =x2y,

wherey =g(x) =3x2

Then the total derivative off with respect to x is

df
dx =
*∂*f
*∂*x +
*∂*f
*∂*y
dy
dx
= 2xy+x2(6x)
= 6x3+6x3
= 12x3

Note: that if we replacedy with g(x)and tool the derivative we would get the same answer – i.e.,

d

De…nition

An antiderivative of a function f(x)is a functionF(x)whose derivative is
the original– i.e., _{dx}d F =f.

De…nition

The function F is also called the inde…nite integral off and written

F(x) =

Z

f(x)dx

R
af(x)dx =aR f(x)dx
R
(f(x) +g(x))dx =R f(x)dx+R g(x)dx
R
xndx = x_{n}n_{+}+_{1}1 +C
R _{1}
xdx =lnx+C
R
ef(x)f0(x)dx =ef(x)+C
R f0(x)
f(x)dx =lnf(x) +C

Z x2+x3+2 x dx = x3 3 + x4 4 +2 ln(x) +C

For …xed numbersa andb, the de…nite integral off(x) fromatob is

F(b) F(a) whereF is the anti-derivative off. Z b

a

f(x)dx =F(b) F(a)

Graphical Interpretation of a De…nite Integral: Area under a curve. Useful when we talk about Consumer and Producer Surplus.

Z 1 0 x2+x3 dx = x 3 3 + x4 4 x=1 x=0 = 1 3 + 1 4 0 3+ 0 4 = 7 12